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arXiv · 2107.13069

Tensor diagrams and cluster combinatorics at punctures

Chris Fraser·Pavlo Pylyavskyy
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Abstract

Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SL(k)-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k is 2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.

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Chris Fraser, Pavlo Pylyavskyy. 2021-07-27. Tensor diagrams and cluster combinatorics at punctures. https://arxiv.org/abs/2107.13069

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